cySomeone pointed out the other day that there are more real numbers between 0 and 1 than there are natural numbers from 1 to infinity, basically by saying if 0.1 = 1, and 0.01 = 2, you can pick a real number between 0 and 1 whose Nth decimal place is 1, for any natural number N. And since 0.2 is <i>not</i> in that series, real numbers between 0 and 1 have <i>at least one</i> more number than all the natural numbers.<br><br>But I was just thinking that it gets even stranger than that. You could also say 0.01 = 1, 0.0001 = 2, where the <code>N*2th</code> decimal place is 1 for every N integers. Another series the size of the set of natural numbers, in between 0 and 1. You don't have to stop there. If the <code>N*3th</code> decimal place is 1, that's another series the size of the natural numbers, and if the <code>N*4th</code> decimal place is 1, that's <i>also</i> a series the size of the natural numbers.<br><br>Now consider that maybe the entire series of 0.1, 0.01, 0.001, 0.0001, etc represents JUST the number 1. The series of 0.01, 0.0001, etc could be 2. If the <code>N*Mth</code> decimal place can be 1, for all natural numbers N, for any natural number M, then that means there are more real numbers between 0 and 1 than there are natural numbers, times natural numbers. That's <code>infinity * infinity</code>.<br><br>I bet a similar proof could also apply to (N,M,O) tuples of 3 natural numbers, and tuples of 4, and so on. That means there are more real numbers between 0 and 1 than there are natural numbers raised to the power of natural numbers. An infinitely exponential number of infinites.<br><br><a href="https://fedicy.us.to?t=math" class="mention hashtag" rel="nofollow noopener" target="_blank">#math</a> <a href="https://fedicy.us.to?t=wat" class="mention hashtag" rel="nofollow noopener" target="_blank">#wat</a> <a href="https://fedicy.us.to?t=rant" class="mention hashtag" rel="nofollow noopener" target="_blank">#rant</a><br>